El argumento del mandato expreso del orden jurídico internacional:

The SMOP algorithm is now applied to a dataset containing the annual January to June steamflow amounts for four rivers in Quebec (Baleine, Churchill Falls, Manicoua- gan and Romaine) from 1972 to 1994. The flow measurements have been recorded in litres per kilometre-squared per second (L/km2_{s). This dataset has previously}

been analysed by Perreault et al. (2000) and was originally published by the Centre d’Expertise Hydrique Quebec. The dataset has been made available in the bcp pack- age (Erdman and Emerson, 2007), from which the data has been obtained. A plot of this data is shown in Figure ??.

5 15 25 35 Baleine 5 15 25 35 Churchill F alls 5 15 25 35 Manicouagan Year 5 15 25 35 Romaine 1972 1974 1976 1978 1980 1982 1984 1986 1988 1990 1992 1994

Figure 4.6.1: The annual January to June streamflow amounts for four rivers in Quebec from 1972 to 1994, measured in L/(km2_s).

Interest lies in detecting changes in the streamflow of the rivers. Whilst Per- reault et al. (2000) search only for shifts in the mean level, visual inspection of the data suggests that changes may be occurring in the mean and/or variance of the flow. Therefore, we consider changes in both properties. Inspection of the series for Churchill Falls may lead to the interpretation that it could be non-stationary near the beginning. If this is believed to be the case, then a non-stationary analysis of this univariate series could be performed, for example using the Locally Stationary Wavelet process (see Nason et al. (2000) for more details). The low-frequency compo- nents could then be filtered out to remove this behaviour and leave the information regarding the mean and variance relatively unaffected. However, in this instance we take the view that this apparent behaviour is simply due to the stochastic nature of the observations, and that the series will be segmented appropriately by a changepoint

CHAPTER 4. MULTIVARIATE CHANGEPOINT DETECTION 96 detection procedure.

Since it is feasible that some rivers may be affected by a change whilst others may not, it is prudent to search for subset-multivariate (rather than strictly fully- multivariate) changes. Therefore, the SMOP algorithm is applied to the data in an effort to detect such changes. To draw further comparisons with the repeated- univariate and fully-multivariate approaches, we apply the univariate PELT algorithm independently to each channel, as well as performing fully-multivariate PELT on the series. For each of the three methods we use a cost function which assumes a Normal likelihood with changes occurring in both mean and variance. For SMOP, we set penalty values α = 2 log n and β = 2 log p log n. For these values of α and β, repeated-univariate PELT is applied with a variable-specific penalty of α + 1

pβ, and

fully-multivariate PELT is applied with penalty pα + β. These penalty choices are made for similar reasons to those discussed in Section 4.5.

The results of applying SMOP, repeated-univariate PELT and fully-multivariate PELT to these Quebec river flows are presented in Figures 4.6.2, 4.6.3(a) and 4.6.3(b) respectively. 5 10 15 20 25 30 35 Baleine 5 10 15 20 25 30 35 Churchill F alls 5 10 15 20 25 30 35 Manicouagan Time 5 10 15 20 25 30 35 Romaine 1972 1973 1974 1975 1976 1977 1978 1979 1980 1981 1982 1983 1984 1985 1986 1987 1988 1989 1990 1991 1992 1993 1994

Figure 4.6.2: The results of applying SMOP to the Quebec river flows. The blue vertical lines represent changepoint locations, and the red horizontal

lines represent the corresponding means of those segments. We see from Figure 4.6.2 that SMOP estimates two changepoints in the series, at the years 1975 and 1984. These two changes affect Churchill Falls and Romaine, and

5 10 15 20 25 30 35 Baleine 5 10 15 20 25 30 35 Churchill F alls 5 10 15 20 25 30 35 Manicouagan Time 5 10 15 20 25 30 35 Romaine 19721973197419751976197719781979198019811982198319841985198619871988198919901991199219931994

4.6.3(a): Repeated-univariate PELT results.

5 10 15 20 25 30 35 Baleine 5 10 15 20 25 30 35 Churchill F alls 5 10 15 20 25 30 35 Manicouagan Time 5 10 15 20 25 30 35 Romaine 19721973197419751976197719781979198019811982198319841985198619871988198919901991199219931994

4.6.3(b): Fully-multivariate PELT results.

Figure 4.6.3: The results of applying repeated-univariate PELT and

fully-multivariate PELT to the Quebec river flows. The blue vertical lines represent changepoint locations, and the red horizontal lines

represent the corresponding means of those segments.

no changepoints are estimated in the river flows of Baleine and Manicouagan. We note that the detected locations correspond to the findings of Perreault et al. (2000), who search for a single changepoint and estimate one at 1984. The multiple changepoint approach of SMOP allows the detection of the additional changepoint.

Comparatively, as can be seen in Figure 4.6.3(a), repeated-univariate PELT also detects a change at 1984, but it detects the change in Baleine, Manicouagan and Romaine, and not Churchill Falls. In addition, the method does not detect a change at 1975 in Churchill Falls or Romaine, and instead detects additional changepoints at varying locations in the flows of the four rivers. These differing locations of changes in the rivers compared to those detected by SMOP is due to the lack of a multivariate consideration, and so multivariate power cannot be harnessed across the four series. Hence, the changes are detected independently.

Similar to SMOP, fully-multivariate PELT detects a changepoint at 1984, but due to the fully-multivariate assumption the change is detected across all rivers. A change- point is also detected at 1976 across all rivers. This is near to the 1975 changepoint detected by SMOP, but has likely been placed slightly different by fully-multivariate PELT due to the necessity of estimating the changepoints in all variables.

CHAPTER 4. MULTIVARIATE CHANGEPOINT DETECTION 98 multivariate PELT reflect the results of Scenario 5 from the simulation study in Sec- tion 4.5. Repeated-univariate PELT seems to overestimate the number of change- points (which can lead to poor estimation of the true change locations), and fully- multivariate PELT generally estimates the correct locations but overestimates the number of affected variables (which, if severe, could begin to affect the location esti- mates).

Given the positive results of SMOP in this applied context, the next section gives consideration to techniques which have the potential to reduce the computational cost of the procedure.